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Step * 2 1 of Lemma adjacent-compatible-iff


1. [X] : CubicalSet
2. I : Cname List
3. L : I-face(X;I) List
4. ∀f1,f2:I-face(X;I).
     ((f1 ∈ L)
     ⇒ (f2 ∈ L)
     ⇒ (¬(dimension(f1) = dimension(f2) ∈ Cname))
     ⇒ ((dimension(f2):=direction(f2))(cube(f1))
        = (dimension(f1):=direction(f1))(cube(f2))
        ∈ X(I-[dimension(f1); dimension(f2)])))
5. i : ℕ||L||
6. j : ℕi
⊢ face-compatible(X;I;L[j];L[i])
BY
{ (Decide ⌜dimension(L[j]) = dimension(L[i]) ∈ Cname⌝⋅ THENA Auto) }

1
1. [X] : CubicalSet
2. I : Cname List
3. L : I-face(X;I) List
4. ∀f1,f2:I-face(X;I).
     ((f1 ∈ L)
     ⇒ (f2 ∈ L)
     ⇒ (¬(dimension(f1) = dimension(f2) ∈ Cname))
     ⇒ ((dimension(f2):=direction(f2))(cube(f1))
        = (dimension(f1):=direction(f1))(cube(f2))
        ∈ X(I-[dimension(f1); dimension(f2)])))
5. i : ℕ||L||
6. j : ℕi
7. dimension(L[j]) = dimension(L[i]) ∈ Cname
⊢ face-compatible(X;I;L[j];L[i])

2
1. [X] : CubicalSet
2. I : Cname List
3. L : I-face(X;I) List
4. ∀f1,f2:I-face(X;I).
     ((f1 ∈ L)
     ⇒ (f2 ∈ L)
     ⇒ (¬(dimension(f1) = dimension(f2) ∈ Cname))
     ⇒ ((dimension(f2):=direction(f2))(cube(f1))
        = (dimension(f1):=direction(f1))(cube(f2))
        ∈ X(I-[dimension(f1); dimension(f2)])))
5. i : ℕ||L||
6. j : ℕi
7. ¬(dimension(L[j]) = dimension(L[i]) ∈ Cname)
⊢ face-compatible(X;I;L[j];L[i])

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Latex:

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Latex:

1.  [X]  :  CubicalSet
2.  I  :  Cname  List
3.  L  :  I-face(X;I)  List
4.  \mforall{}f1,f2:I-face(X;I).
          ((f1  \mmember{}  L)
          {}\mRightarrow{}  (f2  \mmember{}  L)
          {}\mRightarrow{}  (\mneg{}(dimension(f1)  =  dimension(f2)))
          {}\mRightarrow{}  ((dimension(f2):=direction(f2))(cube(f1))  =  (dimension(f1):=direction(f1))(cube(f2))))
5.  i  :  \mBbbN{}||L||
6.  j  :  \mBbbN{}i
\mvdash{}  face-compatible(X;I;L[j];L[i])


By

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Latex:
(Decide  \mkleeneopen{}dimension(L[j])  =  dimension(L[i])\mkleeneclose{}\mcdot{}  THENA  Auto)

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